Characterization of Matrix $K$-Positivity Preserver for $K={\mathbb{R}}^n$ and for Compact Sets $K\subseteq {\mathbb{R}}^n$

Abstract

For any closed $K\subseteq {\mathbb{R}}^n$, in [P.\ J.\ di\,Dio, K.\ Schmüdgen: $K$-Positivity Preserver and their Generators, SIAM J.\ Appl.\ Algebra Geom.\ 9 (2025), 794--824] all $K$-positivity preserver have been characterized, i.e., all linear maps $T:\mathbb{R}[x_1,\dots ,x_n]\to \mathbb{R}[x_1,\dots ,x_n]$ such that $Tp\ge 0$ on $K$ for all $p\ge 0$ on $K$. An important extension of polynomials $\mathbb{R}[x_1,\dots ,x_n]$ with real coefficients are polynomials ${\mathbb{R}}^{m\times m}[x_1,\dots ,x_n]$ with matrix coefficients. Non-negativity on $K$ for matrix polynomials with Hermitian coefficients ${\mathrm{Herm}}_m$ is then $p(x)⪰0$ for all $x\in K$. In the current work, we investigate linear maps $T:{\mathrm{Herm}}_m[x_1,\dots ,x_n]\to {\mathrm{Herm}}_m[x_1,\dots ,x_n]$. We focus on matrix $K$-positivity preserver, i.e., $Tp⪰0$ on $K$ for all $p⪰0$ on $K$. For $K={\mathbb{R}}^n$ and compact sets $K\subseteq {\mathrm{R}}^n$, we give characterizations of matrix $K$-positivity preservers. We discuss the difference between the real and the matrix coefficient case and where our proof fails for general sets $K\subseteq {\mathbb{R}}^n$ with $K\ne {\mathbb{R}}^n$ and $K$ non-compact.